Error Estimates for Solutions of the Semilinear Parabolic Equation in Whole Space

This paper is focused on the error estimates for solutions of the three-dimensional semilinear parabolic equation with initial datau0∈L2(ℝ3). Employing the energy methods and Fourier analysis technique, it is proved that the error between the solution of the semilinear parabolic equation and that of linear heat equation has the behavior asO((1+t)−3/8).

AlgebraicL2Decay for Weak Solutions of the Nonlinear Heat Equations in Whole SpaceR3

We obtained the algebraicL2time decay rate for weak solutions of the nonlinear heat equations with the nonlinear term∇u2uin whole spaceR3. The methods are based on energy methods and Fourier analysis technique.

Energy Partition of Harmonic Waves through a Micro Gap

Energy partition of harmonic waves through a micro gap considering frictional contact was investigated. The incident waves were assumed to be strong enough to make the two solids come into contact. Using the Fourier analysis technique and corrective solution superposition method, the nonlinear boundary problem was reduced to a set of algebraic equations. Special examples for identical materials were discussed in detail. The locations and extents of the separation, slip and stick zones and the energy partition are displayed. Moreover, the effects of gap width, coefficients of friction and incident angle are discussed. These relations are quite effective on seismology and civil engineering.

Influences of an Initial Micro Gap on Wave Transmission

The propagation of elastic waves through a micro gap considering frictional contact is investigated in the present paper. Using the Fourier analysis technique and corrective solution superposition method, the problem is solved analytically. Special examples for identical materials are discussed in detail. The results show that the initial gap width has significant influences on the harmonics and energy partition and their relations may be effective on non-destructive evaluation of a pre-existing gap.

An In Situ XRD Technique For Annealing Investigations

An in situ XRD technique employing a diffractometer equipped with a high temperature camera was used to investigate the annealing behavior of nanoerystalline copper powder produced by mechanical milling. Specimens were annealed isothermally for 12 h at temperatures between 480 and 770 K. The diffraction data was analyzed using a single-profile Fourier analysis technique. The activation energy for diffracting particle growth was determined to be 0.45 eV/atom.